Institut für Numerische Mathematik und Optimierung Probabilistic UQ using PDEs with Random Data: A Case Study Oliver Ernst December 12, 2011 RICAM Special Semester on Multiscale Simulation & Analysis in Energy and the Environment Workshop 4: Multiscale Problems & Stochastic Modelling
Collaborators, Acknowledgments Ingolf Busch, Björn Sprungk TU Freiberg, Institute of Numerical Analysis and Optimization Gerald van den Boogaart TU Freiberg, Institute of Stochastics K. Andrew Cliffe U of Nottingham, School of Mathematical Sciences Elisabeth Ullmann U of Bath, Department of Mathematical Sciences Oliver Ernst (TU Freiberg) 1 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Uncertainty Quantification (UQ) Goal: Account for the fact that numerical simulations in science and engineering are typically based on data which is uncertain. Enabling factors: mature, sophisticated numerical software, highly parallel computing hardware, growing abundance of data. Here: uncertainty modeled as probabilistic. (There are other ways.) Leads to elliptic BVP with random data. Case study: groundwater flow at WIPP Compare 3 different UQ methods for obtaining quantity of interest (contaminant particle travel time). Emphasis on probabilistic model. Oliver Ernst (TU Freiberg) 2 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Uncertainty Quantification (UQ) Goal: Account for the fact that numerical simulations in science and engineering are typically based on data which is uncertain. Enabling factors: mature, sophisticated numerical software, highly parallel computing hardware, growing abundance of data. Here: uncertainty modeled as probabilistic. (There are other ways.) Leads to elliptic BVP with random data. Case study: groundwater flow at WIPP Compare 3 different UQ methods for obtaining quantity of interest (contaminant particle travel time). Emphasis on probabilistic model. Oliver Ernst (TU Freiberg) 2 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Uncertainty Quantification (UQ) Goal: Account for the fact that numerical simulations in science and engineering are typically based on data which is uncertain. Enabling factors: mature, sophisticated numerical software, highly parallel computing hardware, growing abundance of data. Here: uncertainty modeled as probabilistic. (There are other ways.) Leads to elliptic BVP with random data. Case study: groundwater flow at WIPP Compare 3 different UQ methods for obtaining quantity of interest (contaminant particle travel time). Emphasis on probabilistic model. Oliver Ernst (TU Freiberg) 2 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Uncertainty Quantification (UQ) Goal: Account for the fact that numerical simulations in science and engineering are typically based on data which is uncertain. Enabling factors: mature, sophisticated numerical software, highly parallel computing hardware, growing abundance of data. Here: uncertainty modeled as probabilistic. (There are other ways.) Leads to elliptic BVP with random data. Case study: groundwater flow at WIPP Compare 3 different UQ methods for obtaining quantity of interest (contaminant particle travel time). Emphasis on probabilistic model. Oliver Ernst (TU Freiberg) 2 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Outline WIPP Groundwater Flow Problem Parametrization of Random Fields Solution methods for PDEs with Random Data Numerical Results Concluding Remarks Oliver Ernst (TU Freiberg) 3 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Outline WIPP Groundwater Flow Problem Parametrization of Random Fields Solution methods for PDEs with Random Data Numerical Results Concluding Remarks Oliver Ernst (TU Freiberg) 4 UQ via PDEs with Random Data: A Case Study Linz, December 2011
WIPP - Waste Isolation Pilot Plant US DOE repository for radioactive waste situated in New Mexico. Fully operational since 1999. Extensive site characterization and performance assessment since 1976; certification/recertification by US EPA (every 5 years). Large amount of publicly available data. Oliver Ernst (TU Freiberg) 5 UQ via PDEs with Random Data: A Case Study Linz, December 2011
WIPP Groundwater Flow Problem Repository located at a depth of 655m within bedded evaporites, primarily halite (salt). The most transmissive rock in the region is the Culebra Dolomite. In the event of an accidental breach, Culebra would be the principal pathway for transport of radionuclides away from the repository. Oliver Ernst (TU Freiberg) 6 UQ via PDEs with Random Data: A Case Study Linz, December 2011
WIPP Scenario Uncertainty Quantification Problem Release of radionuclides by means of a borehole drilled into the repository. Transport of radionuclides by groundwater through the Culebra Dolomite. Travel time from release point in the repository to the boundary of the region is an important quantity. Flow is two-dimensional to a good approximation. UQ Problem: permeablilty properties of subsurface available only at few observation points. Approach: Model lack of information stochastically, propagate random input data to travel time. Oliver Ernst (TU Freiberg) 7 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Groundwater Flow Model Stationary Darcy flow q = − K ∇ p q : Darcy flux K : hydraulic conductivity p : hydraulic head mass conservation ∇· u = 0 u : pore velocity q = φ u φ : porosity transmissivity T = Kb b : aquifer thickness x ( t ) = − T ( x ) ˙ particle transport bφ ∇ p ( x ) x : particle position x (0) = x 0 x 0 : release location Quantity of interest: particle travel time to reach WIPP boundary (actually, its log ). Oliver Ernst (TU Freiberg) 8 UQ via PDEs with Random Data: A Case Study Linz, December 2011
PDE with Random Coefficient Primal form of Darcy equations: ∇· [ T ( x ) ∇ p ( x )] = 0 , x ∈ D, p = p 0 along ∂D. Model T as a random field (RF) T = T ( x , ω ) , ω ∈ Ω , x ∈ D with respect to underlying probability space (Ω , A , P ) . Assumptions: T has finite mean and covariance T ( x ) = E [ T ( x , · )] , x ∈ D, �� � � �� Cov T ( x , y ) = E T ( x , · ) − T ( x ) T ( y , · ) − T ( y ) , x , y ∈ D. T is lognormal, i.e., Z ( x , ω ) := log T ( x , ω ) is a Gaussian RF. Cov Z is stationary, isotropic, and of Matérn type. Oliver Ernst (TU Freiberg) 9 UQ via PDEs with Random Data: A Case Study Linz, December 2011
WIPP Data 6 x 10 above median transmissivity measurements at 3.595 below median WIPP site boundary 38 test wells head measurements, used to 3.59 obtain boundary data via statistical interpolation 3.585 UTM Northing (kriging) constant layer thickness of 3.58 b = 8 m constant porosity of φ = 0 . 16 3.575 SANDIA Nat. Labs reports [Caufman et al., 1990] 3.57 [La Venue et al., 1990] 6.05 6.1 6.15 6.2 UTM Easting 5 x 10 Oliver Ernst (TU Freiberg) 10 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Outline WIPP Groundwater Flow Problem Parametrization of Random Fields Solution methods for PDEs with Random Data Numerical Results Concluding Remarks Oliver Ernst (TU Freiberg) 11 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Parametrization of Random Fields Merge transmissivity data with statistical model Given: Measurements of transmissivity Assumption: Matérn covariance structure (contains parameters) Regression model Z ( x ) = � k i =1 β i f i ( x ) for mean (incorporate geological information) Procedure: (1) { f i } k i =1 yield point estimates of parameters in Matérn covariance function via restricted maximum likelihood estimation (RML). (2) Condition RF Z = log T on transmissivity measurements. (for Gauss RF coincides with kriging, BLUP, leads to low-rank modification of covariance operator.) (3) Mean of conditioned field is kriging interpolant. (4) Approximate log T by truncated Karhunen-Loève expansion. Oliver Ernst (TU Freiberg) 12 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Parametrization of Random Fields Covariance operator Covariance function of RF Z ∈ L ∞ ( D ) ⊗ L 2 P (Ω) �� �� �� c ( x , y ) = Cov Z ( x , y ) := E Z ( x , · ) − Z ( x ) Z ( y , · ) − Z ( y ) symmetric in x , y ∈ D , positive semidefinite, and continuous on D × D if continuous along ‘diagonal’ { ( x , x ) : x ∈ D } . The covariance operator � C = C Z : L 2 ( D ) → L 2 ( D ) , ( Cu )( x ) = u ( y ) c ( x , y ) d y D is therefore selfadjoint, compact, nonnegative. Its eigenvalues { λ m } m ∈ N form a nonincreasing sequence accumulating at most at 0 . Oliver Ernst (TU Freiberg) 13 UQ via PDEs with Random Data: A Case Study Linz, December 2011
Parametrization of Random Fields Karhunen-Loève expansion Exists sequence of RV { ξ m } m ∈ N ⊂ L 2 P (Ω) , E [ ξ m ] = 0 , E [ ξ k ξ m ] = δ k,m , such that expansion in eigenfunctions { Z m } m ∈ N ∞ � � Z ( x , ω ) = Z ( x ) + λ m Z m ( x ) ξ m ( ω ) m =1 converges in L ∞ ( D ) ⊗ L 2 P (Ω) . [Karhunen, 1947] , [Loève, 1948] Oliver Ernst (TU Freiberg) 14 UQ via PDEs with Random Data: A Case Study Linz, December 2011
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